Question 8
Let $k$ be the incircle of triangle $A B C$ with incentre $I$. The circle $k$ touches the sides $B C$, $C A$ and $A B$ at the points $D$, $E$ and $F$, respectively. Let $G$ be the intersection point of the line $A I$ and the circle $k$ that lies between $A$ and $I$. Assume that $B E$ and $F G$ are parallel. Show that $B D=E F$.